Cavity ring-down (CRD) spectroscopy is an established method for sensitive absorption spectroscopic measurements in gaseous and liquid samples. Extremely sensitive measurements on liquid samples can be performed by filling a conventional cavity with liquid sample, but for many applications in analytical spectroscopy a smaller sample volume is preferred. A cavity of microliter dimensions may be made by reducing the distance between the cavity mirrors to millimeters. Alternatively, one can insert either a cuvette or flow cell, or a liquid film into a larger two-mirror cavity. Another possibility is to use total internal reflection as a third cavity mirror and probe the sample with the evanescent wave. Even smaller volumes of less than 1 μL can be interrogated by using a fiber optic waveguide as the cavity medium.
CRD spectroscopy (CRDS) using a fiber loop is similar to mirror-based CRDS in that the measured ring-down time describes the losses of the fiber-loop cavity containing the sample. While the loss term of interest is the absorption due to the sample, in a fiber cavity there are additional losses due to absorption in the optical fiber, αfiber, and losses at splices and at the sample gap. The ring-down time of a fiber loop with length L, sample gap width d and round trip time tRT is given by:
                    τ        =                                            t              RT                        losses                    =                      nL                          c              ⁡                              (                                                      -                                          ln                      ⁡                                              (                                                  T                                                      gap                            ,                            splice                                                                          )                                                                              +                                                            α                      fiber                                        ⁢                    L                                    +                                      C                    ⁢                                          ɛ                      ~                                        ⁢                    d                                                  )                                                                        (        1        )            The term −ln(Tgap,splice) describes the losses per roundtrip due to the sample gap and fiber splices, C is the concentration of the sample, {tilde over (ε)} is the molar extinction coefficient of the sample based on the natural logarithm (related to the decadic extinction coefficient by {tilde over (ε)}=ε·ln(10)), c is the vacuum speed of light and n is the effective refractive index of the propagating modes.
There are several ways of determining the ring-down time. For example, in pulsed or in continuous wave (cw) CRD spectroscopy the decay of the light intensity is monitored as a function of time. In phase-shift CRD spectroscopy the intensity of the laser is modulated sinusoidally and the cavity emits light that is phase-shifted due to the lifetime of the photons in the cavity, i.e. the ring-down time. The ring-down time can then be obtained by measuring the phase-shift Δφ between the light entering and exiting the cavity at modulation frequency ω=2πν:tan(Δφ)=−ωτ  (2)
For a single-exponential decay there is a linear relationship between the ring-down time τ, the tangent of the phase-shift Δφ and the modulation frequency ω (see equation 2). For multi-exponential decays the relationship between tan Δφ and τ is no longer linear:
                                                                                          tan                  ⁡                                      (                    Δϕ                    )                                                  =                                ⁢                                                      -                    ω                                    ⁢                                                                                                              ∑                                                      j                            =                            1                                                    N                                                ⁢                                                                                                            α                              j                                                        ⁢                                                          τ                              j                              2                                                                                                                                                                          ω                                2                                                            ⁢                                                              τ                                j                                2                                                                                      +                            1                                                                                                                                                ∑                                                      j                            =                            1                                                    N                                                ⁢                                                                                                            α                              j                                                        ⁢                                                          τ                              j                                                                                                                                                                          ω                                2                                                            ⁢                                                              τ                                j                                2                                                                                      +                            1                                                                                                                ⁢                                          ⟶                                              N                        =                        2                                                              ⁢                                          tan                      ⁡                                              (                                                  Δ                          ⁢                                                                                                          ⁢                          ϕ                                                )                                                                                                                                                                                      =                                    ⁢                                                            -                      ω                                        ⁢                                                                                                                        τ                            1                            2                                                                                                                                              ω                                2                                                            ⁢                                                              τ                                1                                2                                                                                      +                            1                                                                          +                                                                                                            α                                                              2                                ,                                1                                                                                      ⁢                                                          τ                              2                              2                                                                                                                                                                          ω                                2                                                            ⁢                                                              τ                                2                                2                                                                                      +                            1                                                                                                                                                                            τ                            1                                                                                                                                              ω                                2                                                            ⁢                                                              τ                                1                                2                                                                                      +                            1                                                                          +                                                                                                            α                                                              2                                ,                                1                                                                                      ⁢                                                          τ                              2                                                                                                                                                                          ω                                2                                                            ⁢                                                              τ                                2                                2                                                                                      +                            1                                                                                                                                              ,                                                    ⁢                                  ⁢                              α                          2              ,              1                                =                                    α              2                                      α              1                                                          (        3        )            The ring-down times can nevertheless be determined by measuring the phase-shift at several modulation frequencies. In optical fibers bi-exponential or tri-exponential decays are observed frequently, since the light is traveling not only in the core but also in the cladding and coating.